Humans and other animals have an evolved ability to detect discrete magnitudes in their environment. Does this observation support evolutionary debunking arguments against mathematical realism, as has been recently argued by Clarke-Doane, or does it bolster mathematical realism, as authors such as Joyce and Sinnott-Armstrong have assumed? To find out, we need to pay closer attention to the features of evolved numerical cognition. I provide a detailed examination of the functional properties of evolved numerical cognition, and (...) propose that they prima facie favor a realist account of numbers. (shrink)

In recent decades, non-representational approaches to mental phenomena and cognition have been gaining traction in cognitive science and philosophy of mind. In these alternative approach, mental representations either lose their central status or, in its most radical form, are banned completely. While there is growing agreement that non-representational accounts may succeed in explaining some cognitive capacities, there is widespread skepticism about the possibility of giving non-representational accounts of cognitive capacities such as memory, imagination or abstract thought. In this paper, (...) I will critically examine the view that there are fundamental limitations to non-representational explanations of cognition. Rather than challenging these arguments on general grounds, I will examine a set of human cognitive capacities that are generally thought to fall outside the scope of non-representational accounts, i.e. numerical cognition. After criticizing standard representational accounts of numerical cognition for their lack of explanatory power, I will argue that a non-representational approach that is inspired by radical enactivism offers the best hope for developing a genuine naturalistic explanatory account for these cognitive capacities. (shrink)

In this paper I study the development of arithmetical cognition with the focus on metaphorical thinking. In an approach developing on Lakoff and Núñez, I propose one particular conceptual metaphor, the Process → Object Metaphor, as a key element in understanding the development of mathematical thinking.

In 1939, the influential psychophysicist S. S. Stevens proposed definitional distinctions between the terms ‘number,’ ‘numerosity,’ and ‘numerousness.’ Although the definitions he proposed were adopted by syeveral psychophysicists and experimental psychologists in the 1940s and 1950s, they were almost forgotten in the subsequent decades, making room for what has been described as a “terminological chaos” in the field of numerical cognition. In this paper, I review Stevens’s distinctions to help bring order to this alleged chaos and to shed (...) light on two closely related questions: whether it is adequate to speak of a number sense and how philosophers can make sense of the claim by cognitive scientists that numbers are perceptual entities. Moreover, I offer further support to Stevens’s distinction between numerosity and numerousness by showing that they are relational properties that emerge through different relationships between agents and their environments. The final conclusion is that, by adopting Stevens’s distinctions, numbers do not need to be seen as perceptual entities, since the so-called number sense is better described as a sense of numerousness. (shrink)

We agree that the approximate number system truly represents number. We endorse the authors' conclusions on the arguments from confounds, congruency, and imprecision, although we disagree with many claims along the way. Here, we discuss some complications with the meanings that undergird theories in numerical cognition, and with the language we use to communicate those theories.

Many researchers, including Clarke and Beck, describe the human numerical system as unitary. We offer an alternative view – the coexistence of several systems; namely, multiple systems existing in parallel, ready to be activated depending on the task/need. Based on this alternative view, we present an account for the representation of rational numbers.

Numbers are vital to so many areas of life: in science, economics, sports, education, and many aspects of everyday life from infancy onwards. This handbook brings together the different research areas that make up the vibrant field of numerical cognition in one comprehensive and authoritative volume.

In this chapter we consider the theories of embodied cognition and extended mind with respect to the human ability to engage in numerical cognition. Such an enquiry requires first distinguishing between our innate number sense and the sort of numerical reasoning that is unique to humans. We provide anthropological and linguistic research to defend the thesis that places the body at the center of our development of numerical reasoning. We then draw on archaeological research to (...) suggest a rough date for when ancient humans first were able to represent numerical information beyond the body and in enduring material artifacts. We conclude by briefly describing how these capacities for embodied and extended numerical cognition shaped our world. (shrink)

This book presents a philosophical interpretation to numerical cognition based on dual process theories and heuristics. It shows how investigations in cognitive science can shed light on issues traditionally raised by philosophers of mathematics. The analysis will also help readers to better understand the relationship between current neuroscientific research and the philosophical reflection on mathematics. The author seeks to explain the acquisition of mathematical concepts. To accomplish this, he needs to answer two questions. How can the concepts of (...) approximate numerosity become an object of thought that is so accessible to our consciousness? How are these concepts refined and specified in such a way as to become numbers? Unfortunately, there is currently no model that can truly demonstrate the role of language in the development of numerical skills starting from approximate pre-verbal skills. However, the author details a solution to this problem: dual process theories. It is an approach widely used by theorists focusing on reasoning, decision making, social cognition, and consciousness. Here, he applies this approach to the studies on mathematical knowledge. He details the results brought about by psychological and neuroscientific studies conducted on numerical cognition by key neuroscientists. In the process, he develops the foundations of a new, potential philosophical explanation on mathematical knowledge. (shrink)

Numbers are vital to so many areas of life: in science, economics, sports, education, and many aspects of everyday life from infancy onwards. This handbook brings together the different research areas that make up the vibrant field of numerical cognition in one comprehensive and authoritative volume.

Carey has argued that there is a system of core numerical cognition – the analog magnitude system – in which cardinal numbers are explicitly represented in iconic format. While the existence of this system is beyond doubt, this paper aims to show that its representations cannot have the combination of features attributed to them by Carey. According to the argument from abstractness, the representation of the cardinal number of a collection of individuals as such requires the representation of (...) individuals as such, and this in turn requires non-iconic format, from which it is concluded that the explicit representation of the cardinal number of some individuals requires non-iconic representational format. In support of the first premise, an account is given of what approximate cardinal numbers might be, and in support of the second, a direct argument is articulated and defended. Finally, in response to an objection, a second argument for the central thesis is provided. While the discussion is couched in the terms of Carey’s work, the considerations it adduces are perfectly general, and the conclusion should therefore be taken into consideration by all those aiming to characterize the AM system. (shrink)

Number morphology (e.g., singular vs. plural) is a part of the grammar that captures numerical information. Some languages have morphological Number values, which express few (paucal), two (dual), three (trial) and sometimes (possibly) four (quadral). Interestingly, the limit of the attested morphological Number values matches the limit of non‐verbal numerical cognition. The latter is based on two systems, one estimating approximate numerosities and the other computing exact numerosities up to three or four. We compared the literature on (...) non‐verbal number systems with data on Number morphology from 218 languages. Our observations suggest that non‐verbal numerical cognition is reflected as a core part of language. (shrink)

Recently, an associative learning account of cognitive control has been suggested (Verguts & Notebaert, 2009). In this so-called adaptation by binding theory, Hebbian learning of stimulus–stimulus and stimulus–response associations is assumed to drive the adaptation of human behavior. In this study, we evaluated the validity of the adaptation-by-binding account for the case of implicit learning of regularities within a stimulus set (i.e., the frequency of specific unit digit combinations in a two-digit number magnitude comparison task) and their association with a (...) particular response. Our data indicated that participants indeed learned these regularities and adapted their behavior accordingly. In particular, influences of cognitive control were even able to override the numerical distance effect—one of the most robust effects in numerical cognition research. Thus, the general cognitive processes involved in two-digit number magnitude comparison seem much more complex than previously assumed. Multi-digit number magnitude comparison may not be automatic and inflexible but influenced by processes of cognitive control being highly adaptive to stimulus set properties and task demands on multiple levels. (shrink)

Based on the notion that time, space, and number are part of a generalized magnitude system, we assume that the dual-systems approach to temporal cognition also applies to numerical cognition. Referring to theoretical models of the development of numerical concepts, we propose that children's early skills in processing numbers can be described analogously to temporal updating and temporal reasoning.

The goals of the present study were to test whether (and which) basic numerical abilities can be improved with training and whether training effects transfer to improvement in children’s math achievement. The literature is mixed with evidence that does or does not substantiate the efficacy of training basic numerical ability. In the present study, we developed a child-friendly software named ‘123 Bakery’ which includes four training modules; non-symbolic numerosity comparison, non-symbolic numerosity estimation, approximate arithmetic and symbol-to-numerosity mapping. Fifty-six (...) first graders were randomly assigned to either the training or control group. The training group participated in 6 weeks of training (5 times a week, 30 minutes per day). All participants underwent pre- and post- training assessment of their basic numerical processing ability (including numerosity discrimination acuity, symbolic/non-symbolic magnitude estimation, approximate arithmetic, symbol-to-numerosity mapping), overall math achievement and intelligence, 6 weeks apart. The acuity for numerosity discrimination (approximate number sense acuity; ANS acuity) significantly improved after training, but this training effect did not transfer to improvement in symbolic, exact calculation, or any other math ability. We conclude that basic numerical cognition training leads to improvement in ANS acuity, but whether this effect transfers to symbolic math ability remains to be further tested. (shrink)

Rips et al. argue that the construction of math schemas roughly similar to the Dedekind/Peano axioms may be necessary for arriving at arithmetical skills. However, they neglect the neo-Fregean alternative axiomatization of arithmetic, based on Hume's principle. Frege arithmetic is arguably a more plausible start for a top-down approach in the psychological study of mathematical cognition than Peano arithmetic.

Lee and Schwarz suggest that separation is the grounded procedure underlying cleansing effects in different psychological domains. Here, we interpret L&S's account from a hierarchical view of cognition that considers the influence of physical properties and sensorimotor constraints on mental representations. This approach allows theoretical integration and generalization of L&S's account to the domain of formal quantitative reasoning.

The domain of numbers provides a paradigmatic case for investigating interactions of culture, language, and cognition: Numerical competencies are considered a core domain of knowledge, and yet the development of specifically human abilities presupposes cultural and linguistic input by way of counting sequences. These sequences constitute systems with distinct structural properties, the cross-linguistic variability of which has implications for number representation and processing. Such representational effects are scrutinized for two types of verbal numeration systems—general and object-specific ones—that were (...) in parallel use in several Oceanic languages. The analysis indicates that the object-specific systems outperform the general systems with respect to counting and mental arithmetic, largely due to their regular and more compact representation. What these findings reveal on cognitive diversity, how the conjectures involved speak to more general issues in cognitive science, and how the approach taken here might help to bridge the gap between anthropology and other cognitive sciences is discussed in the conclusion. (shrink)

Mathematical cognition has become an interesting case study for wider theories of cognition. Menary :1–20, 2015) argues that arithmetical cognition not only shows that internalist theories of cognition are wrong, but that it also shows that the Hypothesis of Extended Cognition is right. I examine this argument in more detail, to see if arithmetical cognition can support such conclusions. Specifically, I look at how the use of numerals extends our arithmetical abilities from quantity-related innate (...) systems to systems that can deal with exact numbers of arbitrary size. I then argue that the system underlying our grasp of small numbers is an unhelpful case study for Menary; it doesn’t support an argument for externalism over internalism. The system for large numbers, on the other hand, clearly displays important interactions between public numeral systems and our cognitive processes. I argue that the large number system supports an argument against internalist theories of arithmetical cognition, but that one cannot conclude that the Hypothesis of Extended Cognition is correct. In other words, the large number case doesn’t decide between the Hypothesis of Extended Cognition and the Hypothesis of EMbedded Cognition. (shrink)

Nonhuman primates represent the most significant extant species for comparative studies of cognition, including such complex phenomena as numerical competence, among others. Studies of numerical skills in monkeys and apes have a long, though somewhat sparse history, although questions for current empirical studies remain of great interest to several fields, including comparative, developmental, and cognitive psychology; anthropology; ethology; and philosophy, to name a few. In addition to demonstrated similarities in complex information processing, empirical studies of a variety (...) of potential cognitive limitations or constraints have provided insights into similarities and differences across the primate order, and continue to offer theoretical and pragmatic directions for future research. An historical overview of primate numerical studies is presented, as well as a summary of the 17‐year research history, including recent findings, of the Comparative Cognition Project at The Ohio State University Chimpanzee Center. Overall, the archival literature on number‐related skills and counting in nonhuman primates offers important implications for revising our thinking about comparative neuroanatomy, cross‐species (human/ape) cognitive similarities and differences, and the evolution of cognition represented by the primate continuum. (shrink)

The world’s diverse written numeral systems are affected by human cognition; in turn, written numeral systems affect mathematical cognition in social environments. The present study investigates the constraints on graphic numerical notation, treating it neither as a byproduct of lexical numeration, nor a mere adjunct to writing, but as a specific written modality with its own cognitive properties. Constraints do not refute the notion of infinite cultural variability; rather, they recognize the infinity of variability within defined limits, (...) thus transcending the universalist/particularist dichotomy. In place of strictly innatist perspectives on mathematical cognition, a model is proposed that invokes domain-specific and notationally-specific constraints to explain patterns in numerical notations. The analysis of exceptions to cross-cultural generalizations makes the study of near-universals highly productive theoretically. The cross-cultural study of patterns in written numbers thus provides a rich complement to the cognitive analysis of writing systems. (shrink)

The world’s diverse written numeral systems are affected by human cognition; in turn, written numeral systems affect mathematical cognition in social environments. The present study investigates the constraints on graphic numerical notation, treating it neither as a byproduct of lexical numeration, nor a mere adjunct to writing, but as a specific written modality with its own cognitive properties. Constraints do not refute the notion of infinite cultural variability; rather, they recognize the infinity of variability within defined limits, (...) thus transcending the universalist/particularist dichotomy. In place of strictly innatist perspectives on mathematical cognition, a model is proposed that invokes domain-specific and notationally-specific constraints to explain patterns in numerical notations. The analysis of exceptions to cross-cultural generalizations makes the study of near-universals highly productive theoretically. The cross-cultural study of patterns in written numbers thus provides a rich complement to the cognitive analysis of writing systems. (shrink)

The idea that there is a “Number Sense” (Dehaene, 1997) or “Core Knowledge” of number ensconced in a modular processing system (Carey, 2009) has gained popularity as the study of numerical cognition has matured. However, these claims are generally made with little, if any, detailed examination of which modular properties are instantiated in numerical processing. In this article, I aim to rectify this situation by detailing the modular properties on display in numerical cognitive processing. In the (...) process, I review literature from across the cognitive sciences and describe how the evidence reported in these works supports the hypothesis that numerical cognitive processing is modular. I outline the properties that would suffice for deeming a certain processing system a modular processing system. Subsequently, I use behavioral, neuropsychological, philosophical, and anthropological evidence to show that the number module is domain specific, informationally encapsulated, neurally localizable, subject to specific pathological breakdowns, mandatory, fast, and inaccessible at the person level; in other words, I use the evidence to demonstrate that some of our numerical capacity is housed in modular casing. (shrink)

Human languages vary in terms of which meanings they lexicalize, but this variation is constrained. It has been argued that languages are under two competing pressures: the pressure to be simple (e.g., to have a small lexicon) and to allow for informative (i.e., precise) communication, and that which meanings get lexicalized may be explained by languages finding a good way to trade off between these two pressures. However, in certain semantic domains, languages can reach very high levels of informativeness even (...) if they lexicalize very few meanings in that domain. This is due to productive morphosyntax and compositional semantics, which may allow for construction of meanings which are not lexicalized. Consider the semantic domain of natural numbers: many languages lexicalize few natural number meanings as monomorphemic expressions, but can precisely convey very many natural number meanings using morphosyntactically complex numerals. In such semantic domains, lexicon size is not in direct competition with informativeness. What explains which meanings are lexicalized in such semantic domains? We will propose that in such cases, languages need to solve a different kind of trade‐off problem: the trade‐off between the pressure to lexicalize as few meanings as possible (i.e, to minimize lexicon size) and the pressure to produce as morphosyntactically simple utterances as possible (i.e, to minimize average morphosyntactic complexity of utterances). To support this claim, we will present a case study of 128 natural languages' numeral systems, and show computationally that they achieve a near‐optimal trade‐off between lexicon size and average morphosyntactic complexity of numerals. This study in conjunction with previous work on communicative efficiency suggests that languages' lexicons are shaped by a trade‐off between not two but three pressures: be simple, be informative, and minimize average morphosyntactic complexity of utterances. (shrink)